An optimized algorithm for computing Wilcoxon's \(T_n^+\) statistic when \(n\) is small (Q1965999)

The Wilcoxon signed rank test is a well known nonparametric statistical test used for analyzing paired replicates data \((X_k,Y_k), k= 1,\dots,n.\) It is based on Wilcoxon's \(T_n^+\) statistic. When \(n\) is large enough (say, \(n>25\)), then \(P(T_n^+ \leq x)\) is calculated by a normal approximation. But when \(n\) is too small (say, \(n\leq 25\)) this approximation is insufficient and the \(T_n^+\) statistic itself must be used. In this article the author proposes a recursive real-time algorithm which computes this probability when \(n\) is small and thus to replace the use of existing tables for statisticians.